Extended Larché–Cahn framework for reactive Cahn–Hilliard multicomponent systems

At high temperature and pressure, solid diffusion and chemical reactions between rock minerals lead to phase transformations. Chemical transport during uphill diffusion causes phase separation, that is, spinodal decomposition. Thus, to describe the coarsening kinetics of the exsolution microstructure, we derive a thermodynamically consistent continuum theory for the multicomponent Cahn–Hilliard equations while accounting for multiple chemical reactions and neglecting deformations. Our approach considers multiple balances of microforces augmented by multiple component content balance equations within an extended Larché–Cahn framework. As for the Larché–Cahn framework, we incorporate into the theory the Larché–Cahn derivatives with respect to the phase fields and their gradients. We also explain the implications of the resulting constrained gradients of the phase fields in the form of the gradient energy coefficients. Moreover, we derive a configurational balance that includes all the associated configurational fields in agreement with the Larché–Cahn framework. We study phase separation in a three-component system whose microstructural evolution depends upon the reaction–diffusion interactions and to analyze the underlying configurational fields. This simulation portrays the interleaving between the reaction and diffusion processes and how the configurational tractions drive the motion of interfaces

Phase-field gradient theory

We propose a phase-field theory for enriched continua. To generalize classical phase-field models, we derive the phase-field gradient theory based on balances of microforces, microtorques, and mass. We focus on materials where second gradients of the phase field describe long-range interactions. By considering a nontrivial interaction inside the body, described by a boundary-edge microtraction, we characterize the existence of a microhypertraction field, a central aspect of this theory. On surfaces, we define the surface microtraction and the surface-couple microtraction emerging from internal surface interactions. We explicitly account for the lack of smoothness along a curve on surfaces enclosing arbitrary parts of the domain. In these rough areas, internal-edge microtractions appear. We begin our theory by characterizing these tractions. Next, in balancing microforces and microtorques, we arrive at the field equations. Subject to thermodynamic constraints, we develop a general set of constitutive relations for a phase-field model where its free-energy density depends on second gradients of the phase field. A priori, the balance equations are general and independent of constitutive equations, where the thermodynamics constrain the constitutive relations through the free-energy imbalance. To exemplify the usefulness of our theory, we generalize two commonly used phase-field equations. We propose a 'generalized Swift-Hohenberg equation'-a second-grade phase-field equation-and its conserved version, the 'generalized phase-field crystal equation'-a conserved second-grade phase-field equation. Furthermore, we derive the configurational fields arising in this theory. We conclude with the presentation of a comprehensive, thermodynamically consistent set of boundary conditions

Phase-field gradient theory

We propose a phase-field theory for enriched continua. To generalize classical phase-field models, we derive the phase-field gradient theory based on balances of microforces, microtorques, and mass. We focus on materials where second gradients of the phase field describe long-range interactions. By considering a nontrivial interaction inside the body, described by a boundary-edge microtraction, we characterize the existence of a hypermicrotraction field, a central aspect of this theory. On surfaces, we define the surface microtraction and the surface-couple microtraction emerging from internal surface interactions. We explicitly account for the lack of smoothness along a curve on surfaces enclosing arbitrary parts of the domain. In these rough areas, internal-edge microtractions appear. We begin our theory by characterizing these tractions. Next, in balancing microforces and microtorques, we arrive at the field equations. Subject to thermodynamic constraints, we develop a general set of constitutive relations for a phase-field model where its free-energy density depends on second gradients of the phase field. A priori, the balance equations are general and independent of constitutive equations, where the thermodynamics constrain the constitutive relations through the free-energy imbalance. To exemplify the usefulness of our theory, we generalize two commonly used phase-field equations. We propose a ‘generalized Swift–Hohenberg equation’—a second-grade phase-field equation—and its conserved version, the ‘generalized phase-field crystal equation’—a conserved second-grade phase-field equation. Furthermore, we derive the configurational fields arising in this theory. We conclude with the presentation of a comprehensive, thermodynamically consistent set of boundary conditions